Pearson Edexcel International Advanced Level

(a) Use the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials to show that $$\sinh 2x + \cosh 2x \equiv (\cosh x + \sinh x)^2.$$

(b) Hence, or otherwise, solve the equation $$\cosh 2x + \sinh 2x = 5,$$ giving your answer in the form $x = \frac{1}{2}\ln k$, where $k$ is an integer.

(a) Let $I_n = \displaystyle \int_0^{\pi/2} \sin^n x \,\mathrm{d}x$ for $n \ge 0$.
Using integration by parts, show that $$n I_n = (n-1)\,I_{n-2}, \qquad n \ge 2.$$

(b) Hence evaluate $\displaystyle \int_0^{\pi/2} \sin^8 x \,\mathrm{d}x$.

A $3 \times 3$ matrix $\mathbf{M}$ is given by $$\mathbf{M} = \begin{pmatrix} k & 2 & 0 \\ 2 & k & 0 \\ 0 & 0 & 1 \end{pmatrix},$$ where $k$ is a constant.

(a) Find $\det(\mathbf{M})$ in terms of $k$.

(b) Determine the values of $k$ for which $\mathbf{M}$ is singular.

(c) For $k = 5$, find the eigenvalues of $\mathbf{M}$ and a corresponding eigenvector for each eigenvalue.

(a) Prove that $\operatorname{arcosh}\,x = \ln\bigl(x + \sqrt{x^2 - 1}\,\bigr)$ for $x \ge 1$.

(b) Solve the equation $$4\sinh x + 3\cosh x = 5,$$ giving your answer in its simplest logarithmic form.

A curve has equation $y = \frac{2}{3}x^{3/2}$ for $0 \le x \le 3$.

(a) Find the arc length of the curve between $x = 0$ and $x = 3$.

(b) The arc of the curve from $x = 0$ to $x = 3$ is rotated completely about the $y$-axis.
Find the exact area of the surface generated.

An ellipse has equation $$\frac{x^2}{25} + \frac{y^2}{9} = 1.$$

(a) Find the eccentricity of the ellipse and the coordinates of its foci.

(b) Find the equation of the tangent to the ellipse at the point $\displaystyle \bigl(4,\,\frac{9}{5}\bigr)$.
Give your answer in the form $ax + by = c$, where $a$, $b$ and $c$ are integers.

(c) Find the equation of the normal to the ellipse at the point $\displaystyle \bigl(4,\,\frac{9}{5}\bigr)$.
This normal meets the $x$-axis at the point $N$. Find the length $FN$, where $F$ is the focus of the ellipse with positive $x$-coordinate.

The line $L$ has vector equation $$\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}, \qquad t \in \mathbb{R}.$$ The plane $\Pi$ has cartesian equation $$x + 2y + 2z = 13.$$

(a) Find the coordinates of the point $P$ where $L$ meets $\Pi$.

(b) Find the acute angle between $L$ and $\Pi$, giving your answer to the nearest $0.1^{\circ}$.

(c) Find the perpendicular distance from the point $(5,\,0,\,1)$ to the plane $\Pi$.

A hyperbola has equation $$\frac{x^2}{16} - \frac{y^2}{9} = 1.$$

(a) Find the eccentricity of the hyperbola and the equations of its directrices.

(b) Show that the tangent to the hyperbola at the point $P(4\sec t,\,3\tan t)$ has equation $$\frac{x\sec t}{4} - \frac{y\tan t}{3} = 1.$$

(c) This tangent meets the asymptotes of the hyperbola at the points $A$ and $B$.
Show that the area of triangle $OAB$, where $O$ is the origin, is independent of $t$, and find this area.

A symmetric matrix $\mathbf{A}$ is given by $$\mathbf{A} = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}.$$

(a) Find the eigenvalues of $\mathbf{A}$ and a corresponding normalised eigenvector for each eigenvalue.

(b) Find an orthogonal matrix $\mathbf{P}$ such that $\mathbf{P}^{\mathrm{T}}\!\mathbf{A}\mathbf{P}$ is a diagonal matrix. State the diagonal matrix $\mathbf{P}^{\mathrm{T}}\!\mathbf{A}\mathbf{P}$.